Proof That the Union of Two Connected Sets Not Mutually Separated are Connected
Theorem
Let
and
be connected sets which are not mutually separated. Then
is connected.
Proof
In a topological space
are said to be mutually separated if
and
Supposeis disconnected then there exist disjoint nonempty open sets
and
such that
with
or
If
and
then
and
are separated sets - a contradiction.
Thus either
or
Therefore
is not a disconnection ofandis connected.